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Properties

Label 2-168-56.3-c1-0-4
Degree $2$
Conductor $168$
Sign $0.881 + 0.471i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.765 − 1.18i)2-s + (−0.866 − 0.5i)3-s + (−0.829 + 1.82i)4-s + (1.61 + 2.79i)5-s + (0.0679 + 1.41i)6-s + (1.82 − 1.91i)7-s + (2.79 − 0.406i)8-s + (0.499 + 0.866i)9-s + (2.08 − 4.05i)10-s + (−1.10 + 1.91i)11-s + (1.62 − 1.16i)12-s + 5.08·13-s + (−3.67 − 0.709i)14-s − 3.22i·15-s + (−2.62 − 3.01i)16-s + (−2.73 − 1.57i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.840i)2-s + (−0.499 − 0.288i)3-s + (−0.414 + 0.910i)4-s + (0.721 + 1.25i)5-s + (0.0277 + 0.576i)6-s + (0.690 − 0.723i)7-s + (0.989 − 0.143i)8-s + (0.166 + 0.288i)9-s + (0.660 − 1.28i)10-s + (−0.333 + 0.577i)11-s + (0.469 − 0.335i)12-s + 1.40·13-s + (−0.981 − 0.189i)14-s − 0.833i·15-s + (−0.656 − 0.754i)16-s + (−0.663 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.881 + 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862881 - 0.216362i\)
\(L(\frac12)\) \(\approx\) \(0.862881 - 0.216362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.765 + 1.18i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-1.82 + 1.91i)T \)
good5 \( 1 + (-1.61 - 2.79i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.10 - 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.08T + 13T^{2} \)
17 \( 1 + (2.73 + 1.57i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.93 + 1.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.65 + 1.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.88iT - 29T^{2} \)
31 \( 1 + (1.01 - 1.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.798 - 0.460i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.96iT - 41T^{2} \)
43 \( 1 + 6.68T + 43T^{2} \)
47 \( 1 + (-1.06 - 1.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.12 + 1.80i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.6 + 6.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.34 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.40 - 7.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (-7.82 - 4.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.60 - 5.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.57iT - 83T^{2} \)
89 \( 1 + (6.32 - 3.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.63070280174229893657691606228, −11.16142099909678476994217081368, −10.94956770955400253533629546969, −10.08852635961293709079398560111, −8.801161000493974937788807717860, −7.41608947350037765055555765765, −6.67093032638689193779757240397, −4.91999618863969840880958810356, −3.24801546396605743872641327724, −1.66122061481958067743907575043, 1.38924587952068667682941597031, 4.52420958026014258441722203600, 5.60013939948387039852175779516, 6.10660779280513111240279385604, 8.016566168974895023611594008413, 8.767692146200065868054014188369, 9.518660162827286134526460983060, 10.76448951834187711920805981560, 11.71644255295712518053918577494, 13.19097286036631542391183119036

Graph of the $Z$-function along the critical line